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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo

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Volume 13, Issue 1


Volume 13 (2015)

The Umbral operator and the integration involving generalized Bessel-type functions

Kottakkaran Sooppy Nisar
  • Department of Mathematics, College of Arts and Science, Prince Sattam bin Abdulaziz University, Wadi Aldawaser, Riyadh region, 11991, Saudi Arabia
  • Other articles by this author:
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/ Saiful Rahman Mondal
  • Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa, 31982, Saudi Arabia
  • Other articles by this author:
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/ Praveen Agarwal
  • Department of Mathematics, Anand International College of Engineering, Jaipur-303012, Republic of India
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/ Mujahed Al-Dhaifallah
  • Department of Systems Engineering, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia
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Published Online: 2015-07-07 | DOI: https://doi.org/10.1515/math-2015-0041


The main purpose of this paper is to introduce a class of new integrals involving generalized Bessel functions and generalized Struve functions by using operational method and umbral formalization of Ramanujan master theorem. Their connections with trigonometric functions with several distinct complex arguments are also presented.

Keywords: Umbral operators; Ramanujan master theorem; Generalized Bessel and Struve functions


  • [1] T. Amdeberhan and V. H. Moll, A formula for a quartic integral: a survey of old proofs and some new ones, Ramanujan J. 18 (2009), no. 1, 91–102. Web of ScienceCrossrefGoogle Scholar

  • [2] T. Amdeberhan,O. Espinosa, I. Gonzalez, M. Harrison, V. H. Moll and A. Straub, Ramanujan’s master theorem, Ramanujan J. 29 (2012), no. 1-3, 103–120. Web of ScienceCrossrefGoogle Scholar

  • [3] P. E. Appell and J. Kampé de Férit, Fonctions Hypergeometriques et Polynomes d’Hermite, Gauthier-Villars, Paris, 1926. Google Scholar

  • [4] D. Babusci, G. Dattoli, G. H. E. Duchamp, Góska and K. A. Penson, Definite integrals and operational methods, Appl. Math. Comput. 219 (2012), no. 6, 3017–3021. Web of ScienceGoogle Scholar

  • [5] D. Babusci and G. Dattoli, On Ramanujan Master Theorem, arXiv:1103.3947 Web of ScienceGoogle Scholar

  • [math-ph]. Google Scholar

  • [6] D. Babusci, G. Dattoli, B. Germano, M. R. Martinelli and P.E. Ricci, Integrals of Bessel functions, Appl. Math. Lett. 26 (2013), no. 3, 351–354. CrossrefGoogle Scholar

  • [7] B. C. Berndt, Ramanujan’s notebooks. Part I, Springer, New York, 1985. Google Scholar

  • [8] Á. Baricz, Geometric properties of generalized Bessel functions, Publ. Math. Debrecen 73 (2008), no. 1-2, 155–178. Google Scholar

  • [9] K. Górska, D. Babusci, G. Dattoli, G. H. E. Duchamp and K. A. Penson, The Ramanujan master theorem and its implications for special functions, Appl. Math. Comput. 218 (2012), no. 23, 11466–11471. Web of ScienceGoogle Scholar

  • [10] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, fifth edition, Academic Press, San Diego, CA, 1996. Google Scholar

  • [11] G. H. Hardy, Ramanujan. Twelve lectures on subjects suggested by his life and work, Cambridge Univ. Press, Cambridge, England, 1940. Google Scholar

  • [12] S. R. Mondal and A. Swaminathan, Geometric properties of generalized Bessel functions, Bull. Malays. Math. Sci. Soc. (2) 35 (2012), no. 1, 179–194. Google Scholar

  • [13] R. Mullin and G.-C. Rota. On the foundations of combinatorial theory III. Theory of binomial enumeration. In B. Harris, editor, Graph theory and its applications, Academic Press, 1970, 167–213. Google Scholar

  • [14] G.-C. Rota. The number of partitions of a set. Amer. Math. Monthly, (1964),no. 71, 498–504. CrossrefGoogle Scholar

  • [15] G.-C. Rota. Finite operator calculus. Academic Press, New York, 1975. Google Scholar

  • [16] G.-C. Rota and B.D. Taylor. An introduction to the umbral calculus. In H.M. Srivastava and Th.M. Rassias, editors, Analysis, Geometry and Groups: A Riemann Legacy Volume, Palm Harbor, Hadronic Press, 1993, 513–525. Google Scholar

  • [17] G.-C. Rota and B.D. Taylor. The classical umbral calculus. SIAM J. Math. Anal.,(1994),no 25, 694–711. Google Scholar

  • [18] S. Roman, The Umbral Calculus (Academic Press, INC, Orlando, 1984). Google Scholar

  • [19] H. M. Srivastava and H. L. Manocha, A treatise on generating functions, Ellis Horwood Series: Mathematics and its Applications, Horwood, Chichester, 1984. Google Scholar

  • [20] F. G. Tricomi, Funzioni ipergeometriche confluenti (Italian), Ed. Cremonese, Roma, 1954. Google Scholar

  • [21] N. Yagmur and H. Orhan, Starlikeness and convexity of generalized Struve functions, Abstr. Appl. Anal. 2013, Art. ID 954513, 6 pp. Web of ScienceGoogle Scholar

  • [22] H. W. Gould and A. T. Hopper, Operational formulas connected with two generalizations of Hermite polynomials, Duke Math. J. 29 (1962), 51–63. CrossrefGoogle Scholar

  • [23] G. Maroscia and P. E. Ricci, Hermite-Kampé de Fériet polynomials and solutions of boundary value problems in the half-space, J. Concr. Appl. Math. 3 (2005), no. 1, 9–29. Google Scholar

  • [24] G. Dattoli et al., Evolution operator equations: integration with algebraic and finite-difference methods. Applications to physical problems in classical and quantum mechanics and quantum field theory, Riv. Nuovo Cimento Soc. Ital. Fis. (4) 20 (1997), no. 2, 3–133. Google Scholar

  • [25] G. Dattoli, P. E. Ricci and C. Cesarano, Monumbral polynomials and the associated formalism, Integral Transforms Spec. Funct. 13 (2002), no. 2, 155–162. Google Scholar

  • [26] G. Dattoli, P. E. Ricci and C. Cesarano, Beyond the monomiality: the monumbrality principle, J. Comput. Anal. Appl. 6 (2004), no. 1, 77–83. Google Scholar

  • [27] C. Cesarano, D. Assante, A note on Generalized Bessel Functions, International Journal of Mathematical Models and Methods in Applied Sciences, vol. 7, p. 625–629 Google Scholar

  • [28] G. Dattoli, C. Cesarano and D. Sacchetti, A note on truncated polynomials, Appl. Math. Comput. 134 (2003), no. 2-3, 595–605. CrossrefGoogle Scholar

About the article

Received: 2015-03-17

Accepted: 2015-06-03

Published Online: 2015-07-07

Citation Information: Open Mathematics, Volume 13, Issue 1, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2015-0041.

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©2015 Kottakkaran Sooppy Nisar et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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